cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Francesco Daddi

Francesco Daddi's wiki page.

Francesco Daddi has authored 7 sequences.

A227336 Decimal expansion of Sum_{k>=1} exp(-k^2)/k.

Original entry on oeis.org

3, 7, 7, 0, 7, 8, 4, 2, 5, 3, 5, 3, 7, 4, 2, 9, 4, 5, 5, 0, 5, 4, 4, 2, 1, 6, 1, 3, 7, 0, 5, 4, 2, 2, 0, 4, 4, 0, 5, 0, 4, 5, 1, 9, 1, 7, 0, 5, 6, 7, 8, 7, 0, 1, 0, 4, 7, 6, 2, 2, 8, 9, 9, 8, 0, 4, 2, 2, 7, 2, 5, 9, 4, 8, 8, 8, 0, 0, 8, 2, 3, 1, 1, 6, 3, 3, 8, 0, 1, 1, 5, 5, 2, 0, 9, 6, 1, 9, 8, 1, 6, 7, 5, 9, 8
Offset: 0

Views

Author

Francesco Daddi, Jul 07 2013

Keywords

Examples

			0.3770784253537429455054421613705422044050451917056787010476228998042...

Programs

  • Maple
    sum(exp(-k^2)/k, k=1..infinity); evalf(%, 120);
  • Mathematica
    digits = 105; NSum[1/(E^(k^2)*k), {k, 1, Infinity}, WorkingPrecision -> digits+1] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Jan 28 2014 *)
  • PARI
    suminf(k=1,exp(-k^2)/k) \\ Charles R Greathouse IV, Jul 07 2013

A193744 Partial sum of Perrin numbers.

Original entry on oeis.org

3, 3, 5, 8, 10, 15, 20, 27, 37, 49, 66, 88, 117, 156, 207, 275, 365, 484, 642, 851, 1128, 1495, 1981, 2625, 3478, 4608, 6105, 8088, 10715, 14195, 18805, 24912, 33002, 43719, 57916, 76723, 101637, 134641, 178362, 236280, 313005, 414644, 549287, 727651, 963933, 1276940, 1691586
Offset: 0

Views

Author

Francesco Daddi, Aug 04 2011

Keywords

Examples

			For n=2, a(2)=Perrin(0)+Perrin(1)+Perrin(2)=3+0+2=5.

Links

Crossrefs

Cf. A001608.

Programs

  • Maple
    perrin[0]:=3: perrin[1]:=0: perrin[2]:=2: a[0]:=3: a[1]:=3: a[2]:=5:  for n from 0 to 100 do perrin[n]:=perrin[n-2]+perrin[n-3]: a[n]:=a[n-1]+perrin[n]: end do;
  • Mathematica
    LinearRecurrence[{0, 1, 1}, {3, 0, 2}, {6, 52}] - 2 (* Alonso del Arte, Aug 05 2011, based on Harvey P. Dale's program for A001608 *)
LinearRecurrence[{1, 1, 0, -1},{3, 3, 5, 8},47] (* Ray Chandler, Aug 03 2015 *)

Formula

a(n) = Perrin(n+5)-2.
a(n) = r1^(n+5)+r2^(n+5)+r3^(n+5)-2, where r1, r2, r3 are the three roots of x^3-x-1 = 0.
G.f.: (3 - x^2)/(1 - x^2 - x^3)/(1-x) = (3 - x^2) / (1 - x - x^2 + x^4). a(n) = a(n-1) + a(n-2) - a(n-4) for n > 2. - Franklin T. Adams-Watters, Aug 05 2011

A192937 a(n) = 100*a(n-1) - (n-1) with a(1)=100.

Original entry on oeis.org

100, 9999, 999898, 99989797, 9998979696, 999897969595, 99989796959494, 9998979695949393, 999897969594939292, 99989796959493929191, 9998979695949392919090, 999897969594939291908989
Offset: 1

Views

Author

Francesco Daddi, Aug 02 2011

Keywords

Examples

			For n=2: a(2)=100*a(1)-(2-1)=100*100-1=10000-1=9999.
For n=3: a(3)=100*a(2)-(3-1)=100*9999-2=999900-2=999898.

Links

Programs

  • GAP
    List([1..20], n -> (98*10^(2*n+2) +99*n +1)/9801); # G. C. Greubel, Feb 06 2019
  • Magma
    [n lt 2 select 100 else 100*Self(n-1)-n+1: n in [1..14]];  // Bruno Berselli, Aug 02 2011
  • Maple
    a[1]:=100; for n from 2 to 12 do a[n]:=100*a[n-1]-(n-1); end do;
  • Mathematica
    LinearRecurrence[{102,-201,100}, {100,9999,999898}, 20] (* G. C. Greubel, Feb 06 2019 *)
RecurrenceTable[{a[1]==100,a[n]==100a[n-1]-n+1},a,{n,20}] (* Harvey P. Dale, May 17 2019 *)
  • PARI
    vector(20, n, (98*10^(2*n+2) +99*n +1)/9801) \\ G. C. Greubel, Feb 06 2019
  • Sage
    [(98*10^(2*n+2) +99*n +1)/9801 for n in (1..20)] # G. C. Greubel, Feb 06 2019

Formula

From Bruno Berselli, Aug 02 2011: (Start)
G.f.: x*(100-201*x+100*x^2)/((1-100*x)*(1-x)^2).
a(n) = (9800*100^n+99*n+1)/9801. (End)

A193640 Indices n such that Perrin(n) > r^n where r is the real root of the polynomial x^3-x-1.

Original entry on oeis.org

0, 2, 3, 5, 8, 10, 13, 15, 16, 18, 20, 21, 23, 26, 28, 31, 33, 34, 36, 39, 41, 44, 46, 47, 49, 51, 52, 54, 57, 59, 62, 64, 65, 67, 69, 70, 72, 75, 77, 80, 82, 83, 85, 87, 88, 90, 93, 95, 96, 98, 100, 101, 103, 106, 108, 111, 113, 114, 116, 118, 119, 121, 124
Offset: 0

Views

Author

Francesco Daddi, Aug 02 2011

Keywords

Comments

r is the so-called plastic number (A060006).
Perrin(n) = r^n + s^n + t^n where r (real), s, t are the three roots of x^3-x-1.
Also Perrin(n) is asymptotic to r^n.
To calculate r^n (for n>2) we can observe that: r^n=s(n)*r^2+t(n)*r+u(n) where s(3)=0, t(3)=1, u(3)=1; s(n+1)=t(n), t(n+1)=s(n)+u(n), u(n+1)=s(n).

Examples

			For n=20 Perrin(20) = A001608(20) = 277 > 276.992... = r^20

Crossrefs

Cf. A001608, A060006, A051017.

Programs

  • Mathematica
    lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n], {n, 0, lim}]; p = CoefficientList[Series[(3 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim + 1], p[[#]] > powers[[#]] &] - 1 (* T. D. Noe, Aug 02 2011 *)

A193627 Indices n such that Perrin(n) < r^n where r is the real root of the polynomial x^3-x-1.

Original entry on oeis.org

1, 4, 6, 7, 9, 11, 12, 14, 17, 19, 22, 24, 25, 27, 29, 30, 32, 35, 37, 38, 40, 42, 43, 45, 48, 50, 53, 55, 56, 58, 60, 61, 63, 66, 68, 71, 73, 74, 76, 78, 79, 81, 84, 86, 89, 91, 92, 94, 97, 99, 102, 104, 105, 107, 109, 110, 112, 115, 117, 120, 122, 123, 125
Offset: 1

Views

Author

Francesco Daddi, Aug 01 2011

Keywords

Comments

r is the so-called plastic number (A060006).
Perrin(n) = r^n + s^n + t^n where r (real), s, t are the three roots of x^3-x-1.
Also Perrin(n) is asymptotic to r^n.
To calculate r^n (for n>2) we can observe that: r^n=s(n)*r^2+t(n)*r+u(n) where s(3)=0, t(3)=1, u(3)=1; s(n+1)=t(n), t(n+1)=s(n)+u(n), u(n+1)=s(n). - Francesco Daddi, Aug 02 2011

Examples

			For n=27 Perrin(27) = A001608(27) = 1983 < 1983.044... = r^27

Crossrefs

Cf. A001608, A060006, A051016.

Programs

  • Mathematica
    lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n], {n, 0, lim}]; p = CoefficientList[Series[(3 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim + 1], p[[#]] <= powers[[#]] &] - 1 (* T. D. Noe, Aug 02 2011 *)

A193599 Indices n such that Padovan(n) > R^n/(2*R+3) where R is the only real root of the polynomial x^3-x-1.

Original entry on oeis.org

0, 3, 5, 6, 8, 10, 11, 13, 16, 18, 21, 23, 24, 26, 28, 29, 31, 34, 36, 39, 41, 42, 44, 46, 47, 49, 52, 54, 55, 57, 59, 60, 62, 65, 67, 70, 72, 73, 75, 77, 78, 80, 83, 85, 88, 90, 91, 93, 95, 96, 98, 101, 103, 106, 108, 109, 111, 114, 116, 119, 121, 122, 124
Offset: 0

Views

Author

Francesco Daddi, Jul 31 2011

Keywords

Comments

R is plastic number (A060006).

Examples

			For n=24, Padovan(24) = 151 > 150.99309... = R^24/(2*R+3).

Crossrefs

Cf. A000931, A060006.

Programs

  • Mathematica
    lim = 200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n/(2*R + 3)], {n, 0, lim}]; p = CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]; Select[Range[lim+1], p[[#]] > powers[[#]] &] - 1 (* T. D. Noe, Aug 01 2011 *)

A193600 Indices n such that Padovan(n) < r^n/(2*r+3) where r is the real root of the polynomial x^3-x-1.

Original entry on oeis.org

1, 2, 4, 7, 9, 12, 14, 15, 17, 19, 20, 22, 25, 27, 30, 32, 33, 35, 37, 38, 40, 43, 45, 48, 50, 51, 53, 56, 58, 61, 63, 64, 66, 68, 69, 71, 74, 76, 79, 81, 82, 84, 86, 87, 89, 92, 94, 97, 99, 100, 102, 104, 105, 107, 110, 112, 113, 115, 117, 118, 120, 123
Offset: 1

Views

Author

Francesco Daddi, Jul 31 2011

Keywords

Comments

R is the so-called plastic number (A060006). Padovan(n) = (r^n)/(2r+3) + (s^n)/(2s+3) + (t^n)/(2t+3) where r (real), s, t are the three roots of x^3-x-1. Also Padovan(n) is asymptotic to r^n / (2*r+3).

Examples

			For n=25, Padovan(25) = A000931(25) = 200 < 200.023... = r^25/(2*r+3).

Crossrefs

Cf. A000931, A060006.

Programs

  • Mathematica
    lim=200; R = Solve[x^3 - x - 1 == 0, x][[1, 1, 2]]; powers = Table[Floor[R^n/(2*R + 3)], {n, lim}]; p = Rest[CoefficientList[Series[(1 - x^2)/(1 - x^2 - x^3), {x, 0, lim}], x]]; Select[Range[lim], p[[#]] <= powers[[#]] &] (* T. D. Noe, Aug 01 2011 *)

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